Comparing Projects With Unequal Lives

Why Unequal Project Lives Need Careful Comparison

When mutually exclusive projects have different operating lives, their standalone NPVs may refer to different lengths of service. A larger NPV for one project does not by itself settle which alternative is more attractive over a common decision horizon. The comparison must account for what happens after the shorter-lived project ends, or translate each project’s value to a consistent annual basis.

Two common methods are equivalent annual value (EAV) and the replacement chain. Both can give a useful comparison when the projects provide comparable service and can be repeated on the stated terms. If the projects are one-time opportunities and cannot be repeated, neither method supplies missing future cash flows; the decision horizon and the cash flows available within it must be specified.

Model: Unless stated otherwise, cash flows occur at year-end, the valuation date is time 0, and $i$ is a constant effective annual interest rate. Write $v=1/(1+i)$. A project’s NPV is the time-0 present value of all its cash flows, including its initial cost and any terminal or replacement cash flow. The comparisons below assume that the same rate is appropriate for all cash flows being compared.

Equivalent Annual Value

Equivalent annual value converts a project’s NPV over its own operating life into a level annual amount with the same present value. The annual amounts are treated as payments at the ends of years 1 through $n$, where $n$ is the project life in years. The ordinary-annuity present-value factor is

a_{\overline{n}|}=\sum_{t=1}^{n}v^t=\frac{1-v^n}{i}.

Definition: If a project has NPV $N$ at time 0 and operates for $n$ years, its equivalent annual value is $$ EAV=\frac{N}{a_{\overline{n}|}}. $$ This is the level year-end amount whose present value over $n$ years equals $N$. EAV is a way to express the project’s value on an annual basis; it does not change the project’s actual cash flows.

The conversion follows directly from the annuity present-value equation. If $EAV$ is paid at each year-end for $n$ years, its present value is $EAV\sum_{t=1}^{n}v^t$. Equating that value to the project’s NPV gives $N=EAV\,a_{\overline{n}|}$; dividing by the positive factor $a_{\overline{n}|}$ yields the definition above. The argument applies to positive or negative NPV.

For comparable repeatable projects, EAV expresses value per year of service. If the projects provide the same service and can be replaced on unchanged terms, the greater EAV indicates the greater annualized value at the stated interest rate. Repeatability matters: without it, annualizing a one-time project may not describe any opportunity that is actually available.

Worked Example: Annualizing Two Repeatable Projects

At an effective annual interest rate of 10%, Project P costs $80 at time 0 and pays $50 at the end of each of years 1 and 2. Project Q costs $110 at time 0 and pays $50 at the end of each of years 1, 2, and 3. Assume both projects can be repeated indefinitely on identical terms, provide the same service, and incur each replacement cost at the end of the preceding project’s life. Compare them using EAV.

Method. Calculate each project’s NPV over its own life, then divide by the annuity factor for that life. The assumption of repeatability and equal service makes the annualized values comparable.

For P, $v=1/1.10=0.9090909$, and $v^2=1/1.10^2=0.8264463$. Therefore, $a_{\overline{2}|}=0.9090909+0.8264463=1.7355372$. Its NPV is $N_P=-80+50(1.7355372)=-80+86.77686=6.77686$. Thus, $EAV_P=6.77686/1.7355372=3.9048$, or $3.90 per year to the nearest cent.

For Q, $v^3=1/1.10^3=0.7513148$, so $a_{\overline{3}|}=0.9090909+0.8264463+0.7513148=2.4868520$. Its NPV is $N_Q=-110+50(2.4868520)=-110+124.34260=14.34260$. Therefore, $EAV_Q=14.34260/2.4868520=5.7675$, or $5.77 per year to the nearest cent.

Project Q has the greater EAV, so it ranks higher under the stated repeatability and service assumptions. The method avoids treating the two- and three-year NPVs as though they covered equal service periods. As a check, each EAV multiplied by its own annuity factor recovers its NPV, apart from rounding.

Replacement Chains and a Common Horizon

A replacement chain compares projects over a horizon that is a common multiple of their lives. At each project’s life-end, the project is replaced by an identical new one. This is appropriate only if replacement is feasible and the repeated project has the same cash flows, costs, service, and operating life as the original. All cash flows in every cycle must be included in that cycle’s NPV.

Suppose a project lasts $n$ years and has NPV $N$ when evaluated at the start of a cycle. Over a horizon of $kn$ years, the successive cycles begin at times $0,n,2n,\ldots,(k-1)n$. Each cycle’s NPV must be discounted from its own start back to time 0. Thus the chain NPV is

N_{\text{chain}}=N\left(1+v^n+v^{2n}+\cdots+v^{(k-1)n}\right).

This formula is a consequence of NPV additivity: the first cycle contributes $N$ at time 0, the second contributes $N v^n$, and each later cycle contributes its cycle NPV discounted by the time at which that cycle begins. It is not correct to add the undiscounted cycle NPVs unless the interest rate is zero.

If two lives are different integer numbers of years, their least common multiple gives one possible common horizon. A shorter common horizon may be possible if the decision horizon is explicitly chosen and the projects’ cash flows can be evaluated consistently through it. The common-horizon calculation must include any terminal value or replacement cost at the horizon. A replacement chain is not justified merely because the lives have a convenient common multiple.

Worked Example: A Six-Year Replacement Chain

At an effective annual interest rate of 8%, a repeatable two-year project has NPV $12 per cycle, and a repeatable three-year project has NPV $20 per cycle. The projects provide the same service, can be repeated indefinitely on identical terms, and each cycle’s NPV is measured at its start. Compare the chains over six years.

Method. Six years accommodates three two-year cycles or two three-year cycles. Discount each later cycle’s NPV from its start to time 0.

For the two-year project, $v^2=1/1.08^2=0.8573388$, and $v^4=1/1.08^4=0.7350299$. Its chain NPV is $12(1+v^2+v^4)=12(1+0.8573388+0.7350299)$. The factor is $2.5923687$, so the chain NPV is $12(2.5923687)=31.1084$, or $31.11 to the nearest cent.

For the three-year project, $v^3=1/1.08^3=0.7938322$. Its chain NPV is $20(1+v^3)=20(1+0.7938322)=20(1.7938322)=35.8766$, or $35.88 to the nearest cent.

The three-year project has the greater six-year chain NPV and ranks higher on this common horizon. As a check, the chain includes three cycle-start NPVs for the two-year project (at years 0, 2, and 4) and two for the three-year project (at years 0 and 3); the later contributions are discounted by the appropriate number of years.

Choosing Between EAV and a Common Horizon

EAV and replacement chains are related ways to put repeatable alternatives on a comparable basis, but they answer the comparison in different forms. EAV reports a level annual amount over each project’s own life. A replacement chain reports an NPV over a selected common horizon. When all assumptions are consistent and the projects are repeatable, the methods should produce a compatible ranking.

Method	What it compares	Key assumption or check
Equivalent annual value	Level annual values over each project’s own life	Comparable annual service and a meaningful repeatability basis
Replacement chain	Discounted NPVs of successive cycles over a common horizon	Cycles can be repeated on identical terms; replacement timing and horizon cash flows are handled correctly
Direct finite-horizon comparison	Actual cash flows within a stated decision horizon	The horizon and cash flows after each project’s life are specified rather than assumed

Worked Example: Comparing Annual Values at a Five Percent Rate

At an effective annual interest rate of 5%, repeatable Project R has NPV $15 per two-year cycle, while repeatable Project S has NPV $30 per four-year cycle. Both projects provide the same service and can be repeated on unchanged terms. Compare their EAVs, rounding to the nearest cent.

Method. Divide each cycle NPV by its own annuity factor. Do not use one project’s life for the other project’s conversion.

For R, $v=1/1.05=0.9523810$ and $v^2=1/1.05^2=0.9070295$. Hence $a_{\overline{2}|}=0.9523810+0.9070295=1.8594105$. Then $EAV_R=15/1.8594105=8.0671$, or $8.07 per year.

For S, $v^3=1/1.05^3=0.8638376$ and $v^4=1/1.05^4=0.8227025$. Thus $a_{\overline{4}|}=0.9523810+0.9070295+0.8638376+0.8227025=3.5459506$. Then $EAV_S=30/3.5459506=8.4604$, or $8.46 per year.

Project S has the greater EAV. The check is that $8.0671(1.8594105)$ is approximately 15 and $8.4604(3.5459506)$ is approximately 30. The common mistake avoided here is dividing by an annuity factor based on the other project’s life.

Common Traps and Limits of the Methods

Comparing unlike horizons directly. A four-year NPV and a two-year NPV cover different operating periods. Use an appropriate annual or common-horizon comparison, or evaluate actual cash flows over a stated finite horizon.
Forgetting to discount later cycles. In a chain, a cycle beginning at time $t$ contributes its NPV multiplied by $v^t$, not its full undiscounted NPV.
Assuming repeatability without support. EAV and replacement chains rely on stated assumptions about future availability and unchanged terms. If those assumptions fail, the comparison may not represent the real decision.
Ignoring service differences. A higher annualized value is not enough to establish preference if the projects provide different amounts or types of service and no adjustment is made for that difference.
Omitting end effects. Salvage, disposal, working capital recovery, and replacement outlays belong in the project cash flows when applicable. A common horizon must also treat its endpoint consistently.

For example, an incorrect chain calculation might add three two-year cycle NPVs of $12 without discounting and report $12+12+12=\$36$. At 8%, the corrected six-year value is $12(1+1/1.08^2+1/1.08^4)=\$31.11$, because the second and third cycle NPVs occur at years 2 and 4, not at time 0. The undiscounted sum is appropriate only at a zero interest rate.

Key takeaway: An unequal-life comparison is fair only after the valuation periods and continuation assumptions are made explicit. Use $EAV=N/a_{\overline{n}|}$ for a level annual equivalent, or discount each repeat cycle’s NPV to a stated common horizon. Do not infer repeatability or equal service from the project lives alone.

Check Your Understanding

For each problem, use the stated effective annual rate and assumptions. Round dollar answers to the nearest cent.

At 6%, a repeatable two-year project has NPV $10 per cycle and provides the same service as a competing alternative. Find its EAV. (Answer: $5.45 per year)
At 8%, a repeatable three-year project has NPV $24 per cycle. Find its EAV. (Answer: $9.31 per year)
At 5%, repeatable two-year and four-year projects have NPVs of $9 and $17 per cycle, respectively, and provide the same service. Compare them over a four-year replacement-chain horizon. (Answer: Two-year chain NPV $=9(1+1/1.05^2)=\$17.16$; four-year chain NPV $=\$17.00$; the two-year project ranks higher.)
At 10%, repeatable three-year and five-year projects have NPVs of $18 and $25 per cycle, respectively, provide the same service, and can be replaced on identical terms. Compare them over 15 years. (Answer: Three-year chain NPV $=18(1+v^3+v^6+v^9+v^{12})=\$55.05$; five-year chain NPV $=25(1+v^5+v^{10})=\$50.16$, where $v=1/1.10$; the three-year project ranks higher.)
At 5%, repeatable two-year and three-year projects provide the same service and have NPVs of $11 and $19 per cycle. Compare by EAV. (Answer: Two-year EAV $=11/1.859410=\$5.92$; three-year EAV $=19/2.723248=\$6.98$; the three-year project ranks higher.)

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